Area Between a Curve & x-Axis

The area $A$ bounded by the curve $y = f(x)$, the x-axis, and the vertical lines $x = a$ and $x = b$ is defined by the definite integral of the function over that interval.

If the function $f(x)$ is non-negative ($f(x) \geq 0$) for all $x$ in $[a, b]$, the area is exactly equal to the definite integral: $A = \int_{a}^{b} f(x) \, dx$

This concept treats the area as a sum of infinitely many rectangles, each with a height determined by the function value $f(x)$ and an infinitesimal width $dx$.

Step 1: Identify the Bounds: Determine the interval $[a, b]$. If these are not provided, find the x-intercepts by setting $f(x) = 0$ and solving for $x$.

Step 2: Check for Sign Changes: Determine if the curve crosses the x-axis within the interval. This is critical because regions below the axis yield negative integral values.

Step 3: Set up the Integral: For regions above the axis, use $\int f(x) \, dx$. For regions below, use $-\int f(x) \, dx$ or $|\int f(x) \, dx|$ to ensure the area magnitude is positive.

When a curve lies below the x-axis ($f(x) < 0$), the definite integral will result in a negative value. Since physical area must be positive, the absolute value of the integral is taken.

If a region spans both above and below the axis, the total area is the sum of the absolute values of the individual integrals for each sub-section.

Mathematically, the total area is expressed as: $Area = \int_{a}^{b} |f(x)| \, dx$